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Matthias Beck

Publications and source records attributed to Matthias Beck.

At least 73 records · Page 4Linked to original sources
arXiv2011

Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs

A \emph{Golomb ruler} is a sequence of distinct integers (the \emph{markings} of the ruler) whose pairwise differences are distinct. Golomb rulers can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for \emph{optimal} Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study [g_m(t) := # {\x \in \Z^{m+1} : \, 0 = x_0 < x_1 < ... < x_{m-1} < x_m = t, \text{all} x_j - x_k \text{distinct}},] the number of Golomb rulers with $m+1$ markings and length $t$. Our main result is that $g_m(t)$ is a quasipolynomial in $t$ which satisfies a combinatorial reciprocity theorem: $(-1)^{m-1} g_m(-t)$ equals the number of rulers $\x$ of length $t$ with $m+1$ markings, each counted with its \emph{Golomb multiplicity}, which measures how many combinatorially different Golomb rulers are in a small neighborhood of $\x$. Our reciprocity theorem can be interpreted in terms of certain mixed graphs associated to Golomb rulers; in this language, it is reminiscent of Stanley's reciprocity theorem for chromatic polynomials. Thus in the second part of the paper we develop an analogue of Stanley's theorem to mixed graphs, which connects their chromatic polynomials to acyclic orientations.

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arXiv2011

Symmetrically Constrained Compositions

Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $λ_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$ constraints ${\sum_{i=1}^n a_i λ_{π(i)} \geq 0 : π\in S_n}$. We show how to compute the generating function of these compositions, combining methods from partition theory, permutation statistics, and lattice-point enumeration.

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arXiv2011

Mahonian Partition Identities Via Polyhedral Geometry

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $Ω$ operator to systematically compute generating functions $\sum_{\la \in P} z_1^{\la_1}...z_n^{\la_n}$ for some set $P$ of integer partitions $\la = (\la_1,..., \la_n)$. Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.

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arXiv2011

An Extreme Family of Generalized Frobenius Numbers

We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_0$ that cannot be represented as a nonnegative integral linear combination of these parameters? More generally, what is the largest integer $g_s$ that has exactly $s$ such representations? We illustrate a family of parameters, based on a recent paper by Tripathi, whose generalized Frobenius numbers $g_0, \ g_1, \ g_2, ...$ exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \ g_{\binom{k+2}{k-1}}, ...$ form an arithmetic progression, and any integer larger than $g_{\binom{k+j}{k-1}}$ has at least $\binom{k+j+1}{k-}$ representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.

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arXiv2010

Nowhere-Harmonic Colorings of Graphs

Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.

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arXiv2010

Bernoulli--Dedekind Sums

Let $p_1,p_2,\dots,p_n, a_1,a_2,\dots,a_n \in \N$, $x_1,x_2,\dots,x_n \in \R$, and denote the $k$th periodized Bernoulli polynomial by $\B_k(x)$. We study expressions of the form \[ \sum_{h \bmod{a_k}} \ \prod_{\substack{i=1\\ i\not=k}}^{n} \ \B_{p_i}\left(a_i \frac{h+x_k}{a_k}-x_i\right). \] These \highlight{Bernoulli--Dedekind sums} generalize and unify various arithmetic sums introduced by Dedekind, Apostol, Carlitz, Rademacher, Sczech, Hall--Wilson--Zagier, and others. Generalized Dedekind sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and combinatorial geometry, and algorithmic complexity. We exhibit a reciprocity theorem for the Bernoulli--Dedekind sums, which gives a unifying picture through a simple combinatorial proof.

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arXiv2010

Six Little Squares and How Their Numbers Grow

We find the numbers of $3 \times 3$ magic, semimagic, and magilatin squares, as functions either of the magic sum or of an upper bound on the entries in the square. Our results on magic and semimagic squares differ from previous ones in that we require the entries in the square to be distinct from each other and we derive our results not by \emph{ad hoc} reasoning but from the general geometric and algebraic method of our paper "An enumerative geometry for magic and magilatin labellings". Here we illustrate that method with a detailed analysis of $3\times3$ squares.

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arXiv2009

Enumeration of $4 \times 4$ Magic Squares

A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart's theory of lattice-point enumeration.

math.CO↗
arXiv2009

Grid graphs, Gorenstein polytopes, and domino stackings

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics.

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arXiv2008

On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series

For every positive integer $n$, consider the linear operator $\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1 - t)^{d + 1}} = \sum_{m \geq 0} g(m) t^{m}$, for some polynomial $g(m)$ with rational coefficients, then $\frac{\U_{n}h(t)}{(1- t)^{d + 1}} = \sum_{m \geq 0} g(nm) t^{m}$. We show that there exists a positive integer $n_{d}$, depending only on $d$, such that if $h(t)$ is a polynomial of degree at most $d$ with nonnegative integer coefficients and $h(0) \geq 1$, then for $n \geq n_{d}$, $\U_{n}h(t)$ has simple, real, strictly negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart $δ$-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen--MacCauley graded rings.

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arXiv2008

Root polytopes and growth series of root lattices

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices A_n, C_n, and D_n, and compute their f-and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway-Mallows-Sloane and Baake-Grimm and proved by Conway-Sloane and Bacher-de la Harpe-Venkov.

math.CO↗
arXiv2008

Finite Trigonometric Character Sums Via Discrete Fourier Analysis

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt-Zaharescu's and some new identities by elementary methods of discrete Fourier Analysis.

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arXiv2008

Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra

We study higher-dimensional analogs of the Dedekind-Carlitz polynomials c(u,v;a,b) := sum_{k=1..b-1} u^[ka/b] v^(k-1), where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law (v-1) c(u,v;a,b) + (u-1) c(v,u;b,a) = u^(a-1) v^(b-1) - 1, from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind-Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind-Carlitz polynomials, a characterization of Dedekind-Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell-Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes.

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arXiv2007

Logic Design for On-Chip Test Clock Generation - Implementation Details and Impact on Delay Test Quality

This paper addresses delay test for SOC devices with high frequency clock domains. A logic design for on-chip high-speed clock generation, implemented to avoid expensive test equipment, is described in detail. Techniques for on-chip clock generation, meant to reduce test vector count and to increase test quality, are discussed. ATPG results for the proposed techniques are given.

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arXiv2007

Maximal Periods of (Ehrhart) Quasi-Polynomials

A \emph{quasi-polynomial} is a function defined of the form $q(k) = c_d(k) k^d + c_{d-1}(k) k^{d-1} + ... + c_0(k)$, where $c_0, c_1, ..., c_d$ are periodic functions in $k \in \Z$. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the $c_j(k)$ for Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expected period and present a general theorem that yields maximal periods for the coefficients of certain quasi-polynomials. We present a construction for (Ehrhart) quasi-polynomials that exhibit maximal period behavior and use it to answer a question of Zaslavsky on convolutions of quasi-polynomials.

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arXiv2006

The Number of Nowhere-Zero Flows on Graphs and Signed Graphs

A nowhere-zero $k$-flow on a graph $Γ$ is a mapping from the edges of $Γ$ to the set $\{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ$ such that, in any fixed orientation of $Γ$, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an \emph{integral flow polynomial} that counts nowhere-zero $k$-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.

math.CO↗
arXiv2006

Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices

The coordination sequence of a lattice $Ł$ encodes the word-length function with respect to $M$, a set that generates $Ł$ as a monoid. We investigate the coordination sequence of the cyclotomic lattice $Ł= \Z[ζ_m]$, where $ζ_m$ is a primitive $mþ$ root of unity and where $M$ is the set of all $mþ$ roots of unity. We prove several conjectures by Parker regarding the structure of the rational generating function of the coordination sequence; this structure depends on the prime factorization of $m$. Our methods are based on unimodular triangulations of the $mþ$ cyclotomic polytope, the convex hull of the $m$ roots of unity in $\R^{ϕ(m)}$, with respect to a canonically chosen basis of $Ł$.

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